Question

A bob of mass m is suspended at a point O by a light string of length l and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity \(v_{0}\) at the point 'A'. the string becomes slack when, the bob reaches at the point 'D'. The ratio of the kinetic energy of the bob at the points B and C is ________ .

(1) 2

(2) 1

(3) 4

(4) 3

Answer 1

Correct option is: (1) 2

\(\frac{1}{2} \mathrm{mv}_{\mathrm{A}}^{2}=\frac{1}{2} \mathrm{mv}_{\mathrm{B}}^{2}+\mathrm{mgh}\)

\( \Rightarrow \frac{1}{2} \mathrm{~m}(5 \mathrm{~g} \ell)=\frac{1}{2} \mathrm{mv}_{\mathrm{B}}^{2}+\mathrm{mg} \frac{\ell}{2}\)

\( \Rightarrow \frac{5 \mathrm{mg} \ell}{2}-\frac{\mathrm{mg} \ell}{2}=\mathrm{KE}_{\mathrm{B}}\)

\( \Rightarrow \mathrm{KE}_{\mathrm{B}}=2 \mathrm{mg} \ell\)

\( \frac{1}{2} m v_{\mathrm{C}}^{2}=\frac{1}{2} m v_{\mathrm{D}}^{2}+m g \frac{\ell}{2}\)

\( \Rightarrow \mathrm{KE}_{\mathrm{C}}=\frac{1}{2} \mathrm{mg} \ell+\mathrm{mg} \frac{\ell}{2}=\mathrm{mg} \ell\)

\( \Rightarrow \frac{\mathrm{KE}_{\mathrm{B}}}{\mathrm{KE}_{\mathrm{C}}}=2\)